Theoretical Background
3.1.2 Heat generation model by different tool geometries
Control of heat generation is prerequisite for high quality weld during friction stir welding (FSW) processes. It is important to have an adequate mathematical model capable to precisely describe heat generation during FSW. There are numerous models that explain heat generation and provide results with various degrees of accuracy since these models include numerous approximations and neglect some key parameters. Mathematical model specified in this work describes contact condition, contact pressure, friction, thermal history and points out the dual nature of heat generation process i.e. adhesion and deformation component in total heat generation. It is concluded that adhesion component of total heat dominates than deformation component.
The heat generated at the interface of tool and work piece due to friction is Qf and due to plastic deformation is Qp. Therefore, total elemental heat due to friction and plastic deformation is given by
dQFSW = dQf+ dQp (3.6)
The material flow and heat generation are characterized by the contact conditions at the interface, and are described as sliding, sticking or partial sliding/sticking condition [262].
The contact condition under the shoulder can be described by sliding friction, using a
Chapter 3 friction coefficient μ and interfacial pressure p, or sticking friction, based on the interfacial shear strength at an appropriate temperature and strain rate. It is convenient to define a contact state variable δ, which relates the velocity of the contact workpiece surface with the velocity of the tool surface. The contact state variable (δ) is assumed to change linearly with the distance from the centre of the pin. Based on these assumptions and geometrical aspects of the deformation zone, the contact state variable can be expressed as
δ =Vmatrix
Vtool (3.7)
where Vmatrix is peripheral velocity of workpiece and Vtool is peripheral velocity of tool. A 100% sticking condition at the pin/material interface is assumed i.e. same as tool velocity.
The velocity of the material is equal to zero at the outer edge of the deformation. The contact shear stress is then
τcontact= τy = σy
√3 (3.8)
If the contact shear stress is smaller than the matrix shear yield stress, the matrix segment volume shears slightly to a stationary elastic deformation where the shear stress equals the
‘dynamic’ contact shear stress. This state is referred to as the sliding condition. Coulomb’s friction law is used to describe the critical friction stress necessary for a sliding condition:
τcontact= μp (3.9)
The frictional heat generation on an elemental area dA at the tool-work-piece interface is expressed as [8]
dQf= (1 − δ)ωr𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡dA (3.10)
Theoretical Background
Physically, δ (<1) accounts the amount of frictional work dissipated into the workpiece, ω is angular velocity of tool and r is radial distance of elemental area dA from axis. The heat generated due to plastic shear deformation Qp sticking to the tool is given by [263]
dQp = δωrτcontactdA (3.11)
The last possible state between the sticking and sliding condition is a mixed state. In this case, the matrix segment accelerates to a velocity less than the tool surface velocity where it stabilizes. The equilibrium establishes when the ‘dynamic’ contact shear stress equals the internal yield shear stress due to a quasi-stationary plastic deformation rate. This is referred to as the partial sliding/sticking condition. In this model, there is no difference between the dynamic and the static friction coefficients. Therefore, the total heat generation due to sliding and sticking is expressed as
QFSW = δ ∗ QFSW,sticking+ (1 − δ) ∗ QFSW,sliding (3.12)
Straight cylindrical tool pin
Estimation of the amount of heat generated during FSW is based on analytical expression on active surface of the welding tool. In order to calculate the heat generation in the shoulder surface rotating around the tool centre axis, an infinitesimal segment on that surface is considered. The infinitesimal segment area dA = rdθdr is exposed to a uniform contact shear stress τcontact. This segment contributes with an infinitesimal force of dF = τcontact·dA and torque of dM = r·dF. The angle between shoulder surface and This surface is consider as active surface of tool which takes part in welding process as shown in Fig. 3.3.
Therefore, the heat energy generated at the contact interface between a rotating FSW tool and a stationary workpiece are subdivided as Q1, Q2 and Q3 i.e. on the tool shoulder’s surface, tool pin’s side surface and tool pin’s tip surface, respectively. The amount of translation heat is significantly smaller than amount of rotational heat and it can be neglected in analysis.
Qtotal = Q1 + Q2 + Q3 (3.13)
Chapter 3 Q1 = ∫ ∫RRshoulderωτcontactr2(1 + tan α)
probe 2π
0 drdƟ (3.14)
where 𝛼 is the angle produced by the conical shoulder (Fig. 3.3). Heat generation from the probe are expressed as
Q2 = ∫02π∫0HprobeωτcontactR2probedrdƟ = 2πωτcontactR2probeHprobe (3.15)
Q3 = ∫02π∫0Rprobeωτcontactr2drdƟ =2
3πωτcontactRprobe3 (3.16) Therefore the total heat generation due to sliding or sticking friction is estimated as
Q = 2
3πωτcontact((R3shoulder− R3probe) + R3probe+ 3R2probeHprobe) (3.17) For a flat shoulder and straight cylindrical tool, the total heat generation is a linear combination of sliding and sticking condition and is expressed as
QFSW= 2
3πω[δτy+ (1 − δ)μp]{(R3shoulder− Rprobe3 ) + R3probe+ 3R2probeHprobe}(3.18)
dr dz
Pin side surface Pin bottom surface
Shoulder surface FSW tool
Tool sh oulder Tool pin
Dshoulder
rdƟ
dƟ
dr Dprobe
rdƟ
dr dƟ
Dprobe
Hprobe
dƟ
rdƟ
α Rshoulder
Rprobe
Figure 3.3: Schematic illustration of nomenclature of straight pin tool.
Theoretical Background Straight tapered tool pin
An analytical model for heat generation for friction stir welding using taper cylindrical pin profile is developed. The analytical expression is the modification of previous analytical models known from the literature which is verified and well matches with the model developed by previous researchers. Figure 3.4 describes the typical FSW tool having the geometric shape of a taper cone without any thread along with flat shaped shoulder.
Figure 3.4: Illustration of nomenclature of taper cylindrical pin profile.
The probe consists of a taper cylindrical surface with a bottom radius of Rbottom, top radius Rtop and probe height Hprobe. The heat generated from the probe is given by Eq. (3.21) over the probe side area
Q2 = ∫02𝜋∫ 𝜔𝑟0𝑙 2𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡dθdz (3.19)
Q2 = 2πω𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑙 (𝑅𝑡𝑜𝑝+𝑅𝑏𝑜𝑡𝑡𝑜𝑚
2 )2 (3.20)
Q2 = πω𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡
2
𝐻𝑝𝑟𝑜𝑏𝑒
𝑐𝑜𝑠𝜓 (𝑅𝑡𝑜𝑝+ 𝑅𝑏𝑜𝑡𝑡𝑜𝑚)2 (3.21)
Chapter 3 Different parts of active surfaces of STT pin profile FSW tool result in different amounts of heat generated on them that give different expressions for estimating the amount of generated heat. After the assimilation of Q1, Q2 and Q3, the expressions for the analytical amount of heat generated are, respectively
𝑄𝑡𝑜𝑡𝑎𝑙 = 𝑄1+𝑄2+ 𝑄3 (3.22)
Therefore, the total heat generation due to sliding or sticking friction is estimated as 𝑄𝑇 =2
3𝜋𝜔𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡(𝑅𝑠ℎ𝑜𝑢𝑙𝑑𝑒𝑟3 + 𝑅𝑡𝑜𝑝3 ) + 2πω𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡πω𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡
2
𝐻𝑝𝑟𝑜𝑏𝑒
𝑐𝑜𝑠𝜓 (𝑅𝑡𝑜𝑝 + 𝑅𝑏𝑜𝑡𝑡𝑜𝑚)2+
2
3πω𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑅𝑏𝑜𝑡𝑡𝑜𝑚3 (3.23)